Optimal. Leaf size=42 \[ -\frac{x \coth (a+b x)}{b^2}+\frac{\log (\sinh (a+b x))}{b^3}-\frac{x^2 \text{csch}^2(a+b x)}{2 b} \]
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Rubi [A] time = 0.0581759, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {5419, 4184, 3475} \[ -\frac{x \coth (a+b x)}{b^2}+\frac{\log (\sinh (a+b x))}{b^3}-\frac{x^2 \text{csch}^2(a+b x)}{2 b} \]
Antiderivative was successfully verified.
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Rule 5419
Rule 4184
Rule 3475
Rubi steps
\begin{align*} \int x^2 \coth (a+b x) \text{csch}^2(a+b x) \, dx &=-\frac{x^2 \text{csch}^2(a+b x)}{2 b}+\frac{\int x \text{csch}^2(a+b x) \, dx}{b}\\ &=-\frac{x \coth (a+b x)}{b^2}-\frac{x^2 \text{csch}^2(a+b x)}{2 b}+\frac{\int \coth (a+b x) \, dx}{b^2}\\ &=-\frac{x \coth (a+b x)}{b^2}-\frac{x^2 \text{csch}^2(a+b x)}{2 b}+\frac{\log (\sinh (a+b x))}{b^3}\\ \end{align*}
Mathematica [A] time = 0.114799, size = 55, normalized size = 1.31 \[ -\frac{x \coth (a)}{b^2}+\frac{\log (\sinh (a+b x))}{b^3}+\frac{x \text{csch}(a) \sinh (b x) \text{csch}(a+b x)}{b^2}-\frac{x^2 \text{csch}^2(a+b x)}{2 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.03, size = 72, normalized size = 1.7 \begin{align*} -2\,{\frac{x}{{b}^{2}}}-2\,{\frac{a}{{b}^{3}}}-2\,{\frac{x \left ( bx{{\rm e}^{2\,bx+2\,a}}+{{\rm e}^{2\,bx+2\,a}}-1 \right ) }{{b}^{2} \left ({{\rm e}^{2\,bx+2\,a}}-1 \right ) ^{2}}}+{\frac{\ln \left ({{\rm e}^{2\,bx+2\,a}}-1 \right ) }{{b}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.52887, size = 144, normalized size = 3.43 \begin{align*} -\frac{2 \,{\left ({\left (b x^{2} e^{\left (2 \, a\right )} - x e^{\left (2 \, a\right )}\right )} e^{\left (2 \, b x\right )} + x e^{\left (4 \, b x + 4 \, a\right )}\right )}}{b^{2} e^{\left (4 \, b x + 4 \, a\right )} - 2 \, b^{2} e^{\left (2 \, b x + 2 \, a\right )} + b^{2}} + \frac{\log \left ({\left (e^{\left (b x + a\right )} + 1\right )} e^{\left (-a\right )}\right )}{b^{3}} + \frac{\log \left ({\left (e^{\left (b x + a\right )} - 1\right )} e^{\left (-a\right )}\right )}{b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.46102, size = 979, normalized size = 23.31 \begin{align*} -\frac{2 \, b x \cosh \left (b x + a\right )^{4} + 8 \, b x \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + 2 \, b x \sinh \left (b x + a\right )^{4} + 2 \,{\left (b^{2} x^{2} - b x\right )} \cosh \left (b x + a\right )^{2} + 2 \,{\left (b^{2} x^{2} + 6 \, b x \cosh \left (b x + a\right )^{2} - b x\right )} \sinh \left (b x + a\right )^{2} -{\left (\cosh \left (b x + a\right )^{4} + 4 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + \sinh \left (b x + a\right )^{4} + 2 \,{\left (3 \, \cosh \left (b x + a\right )^{2} - 1\right )} \sinh \left (b x + a\right )^{2} - 2 \, \cosh \left (b x + a\right )^{2} + 4 \,{\left (\cosh \left (b x + a\right )^{3} - \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + 1\right )} \log \left (\frac{2 \, \sinh \left (b x + a\right )}{\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )}\right ) + 4 \,{\left (2 \, b x \cosh \left (b x + a\right )^{3} +{\left (b^{2} x^{2} - b x\right )} \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )}{b^{3} \cosh \left (b x + a\right )^{4} + 4 \, b^{3} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + b^{3} \sinh \left (b x + a\right )^{4} - 2 \, b^{3} \cosh \left (b x + a\right )^{2} + b^{3} + 2 \,{\left (3 \, b^{3} \cosh \left (b x + a\right )^{2} - b^{3}\right )} \sinh \left (b x + a\right )^{2} + 4 \,{\left (b^{3} \cosh \left (b x + a\right )^{3} - b^{3} \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.17487, size = 188, normalized size = 4.48 \begin{align*} -\frac{2 \, b^{2} x^{2} e^{\left (2 \, b x + 2 \, a\right )} + 2 \, b x e^{\left (4 \, b x + 4 \, a\right )} - 2 \, b x e^{\left (2 \, b x + 2 \, a\right )} - e^{\left (4 \, b x + 4 \, a\right )} \log \left (e^{\left (2 \, b x + 2 \, a\right )} - 1\right ) + 2 \, e^{\left (2 \, b x + 2 \, a\right )} \log \left (e^{\left (2 \, b x + 2 \, a\right )} - 1\right ) - \log \left (e^{\left (2 \, b x + 2 \, a\right )} - 1\right )}{b^{3} e^{\left (4 \, b x + 4 \, a\right )} - 2 \, b^{3} e^{\left (2 \, b x + 2 \, a\right )} + b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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